Asymptotes | Horizontal and Verticle
Lets know what is asymptotes with example.
"A straight line l is called an asymptote for a curve c if the distance btween l and c approaches zero as the distance moved along l (from some fixed point on l)tends to infinity."
"The curve c can approach asymptote l as o e moves along l in one direction or in the opposite direction or in both the directions."
"Suppose the equation y=f(x) of c is such that y is real and y-->a as x-->∞ or x-->-∞, then y=a is a horizontal asymptote. For the distance between the curve and the straight line y=a is y-a and this approaches zero as x-->∞ or x-->-∞."
"If the equation of c is such that y is real and y-->∞ or -∞ as x-->a from one side then the straight line x=a is a verticle asymptote. To see this observe that (i)x-a is the distance between the curve and the straight line and that this distance is supposed to approach zero (ii)y-->∞ or -∞ as x-->a so that
Lim = lim (x-a) = 0
Thus to locate vertical asymptotes we have to find a number a such that lim y = ∞ or - ∞."
"Similarly if y-->mx+c as x-->∞ or -∞ then y=mx+c is an asymptote( which is neother vertical nor horizontal."
"Thus we inquire for lim xapproaches to + ,- infinity y in studying asymptote."
"For algebraic equations we can dind horizontal and vertical asymptotes as follows:
For horizontal asymptotes we write the given equation in the form x= Ψ(y)/ θ(y) and consider those values of y for which theta y is 0."
"Similarly to find a vertical asymptote we write the given equation in the form y=f(x)/g(x) and consider those values of x for which g(x)=0."
Working rule for asymptotes parallel to the axes: "In an equation of a curve the coefficient of the highest power of x (respectively of y) equated to zero gives asymptotes (if any) parallel to the x axis (respectively y axis)."
Lets consider an example.
Lets find the asymptote of the curve y=1/(x-2)²
When x-->2 from either side ,y is positive and approaches ∞ . Thus x=2 is a verticle asymptote.
Again lim y = 0 thus y=0 is a horizontal
Thus asymptote can be readily obtained by the working rule stated above.
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